Electrical Power Transmission

Lessons

Senior 3 · Physics

Electrical Power Transmission

4 lessons

Lesson 3 of 4

Transmitting Electrical Power and Reducing Losses

25 min

Why such dangerous voltages?

Rwanda's main transmission lines carry electricity at 110 000 V and 220 000 V across the country, from hydropower, methane, peat and solar plants to the towns that use it (REG). Voltages like these can kill a person who comes within a few metres of a line, so they make pylons tall, insulators long and construction expensive. Why not simply send the electricity at the much lower 230 V that comes out of a socket? The answer lies in a single formula: the heat wasted in a wire is $I^{2} R$ .

Power lost in the lines

Every transmission line has resistance $R$ , so the current $I$ flowing through it heats the wire. The power wasted as heat is

P_{\text{loss}} = I^2 R

To deliver a power $P$ at a voltage $V$ , the line must carry a current $I = \frac{P}{V}$ . So

P_{\text{loss}} = \left(\frac{P}{V}\right)^{2} R

For a given power and line, the loss is proportional to $\frac{1}{V^2}$ : double the voltage and the loss falls to a quarter; multiply the voltage by 10 and the loss falls to one hundredth.

Worked example 1: the same power at different voltages

A power station sends 10 MW along a line of total resistance 5.0 Ω (illustrative values). Compare the power lost when it is sent at 11 kV, 110 kV and 220 kV.

Voltage	Current $I = \frac{P}{V}$	Loss $I^{2} R$	Percentage lost
11 kV	909 A	4.13 MW	41%
110 kV	90.9 A	41.3 kW	0.41%
220 kV	45.5 A	10.3 kW	0.10%

For example, at 110 kV:

I = \frac{10 \times 10^{6}}{110 \times 10^{3}} = 90.9\ \text{A} \qquad P_{\text{loss}} = 90.9^{2} \times 5.0 \approx 41\,300\ \text{W}

At 230 V the line would need a current of more than 43 000 A, and the calculated "loss" would be larger than the power being sent: it simply could not be done.

Worked example 2: how much resistance does a line have?

Each conductor of a 50 km line is aluminium (resistivity $2.8 \times 10^{-8}\ \Omega\,\text{m}$ ) with a cross-sectional area of 400 mm². Find its resistance.

A = 400\ \mathrm{mm^2} = 4.0 \times 10^{-4}\ \mathrm{m^2} \qquad R = \frac{\rho L}{A} = \frac{2.8 \times 10^{-8} \times 50\,000}{4.0 \times 10^{-4}} = 3.5\ \Omega

Aluminium is used rather than copper because it is much lighter and cheaper, although its resistivity is a little higher; the thick conductors keep $R$ small.

Definition

Transmission: carrying electrical energy over long distances from power stations to substations, at high voltage. Distribution: carrying it from substations to consumers at medium and low voltage.

National grid: the network of power stations, transformers, substations, transmission and distribution lines that connects generators to consumers across a country.

From power station to home

Generation: generators in the power stations produce alternating current, typically at around 11 kV.
Step-up: a transformer at the power station raises the voltage for transmission. Rwanda's transmission lines run at 110 kV and 220 kV; by the end of June 2023 they totalled 1,158 km (REG). In January 2023, for example, a 117 km high-voltage line from the Rusumo hydropower plant to the Bugesera substation was energised (REG).
Step-down at substations: transformers lower the voltage to 30 kV for rural lines or 15 kV for some urban lines (REG).
Distribution: medium-voltage lines on poles carry it through towns and villages.
Final step-down: pole or ground-mounted transformers give 400 V and 230 V to homes, schools and businesses (REG).

The whole system uses alternating current, because only a.c. can be stepped up and down by transformers.

Why not an even higher voltage?

Raising the voltage further would cut the $I^{2} R$ loss even more, but every step up costs money and space: the wires must be held further from the pylon and from the ground by longer insulators and taller towers, wider strips of land must be kept clear under the lines, and at very high voltages the air around the wires starts to conduct, wasting energy as a crackling glow (corona). Each country chooses voltages that balance these costs against the savings in lost energy. Thick conductors of low resistance help too, but they are heavy, so the voltage is the main tool.

Warning

Use the current in the line, not the voltage, in $P = I^{2} R$ . A common mistake is to write the loss as $\frac{V^2}{R}$ with the transmission voltage, which gives an enormous (and wrong) answer. The transmission voltage is not across the line's resistance; most of it is across the consumers at the far end. Find $I = \frac{P}{V}$ first, then use $I^{2} R$ .

Tip

High voltage, low current, low loss. For the same power, $P = V I$ : raise $V$ by a factor $k$ and $I$ falls by $k$ , so the loss $I^{2} R$ falls by $k^{2}$ .

The National Grid: how we generate and transmit electricity (Cognito). Watch for the step-up and step-down transformers and why a high voltage means a low current and small losses.

✦

Practice

Explain why electrical power is transmitted at a high voltage.
A power station sends 20 MW along a line of resistance 4.0 Ω. Find the power lost in the line if the power is sent at (a) 110 kV; (b) 220 kV.
A 2000 W heater is used at 230 V at the end of a long extension cable of total resistance 0.50 Ω. Find the current and the power wasted in the cable.
An aluminium transmission conductor is 100 km long with a cross-section of 500 mm². Find its resistance ( $\rho = 2.8 \times 10^{-8}\ \Omega\,\text{m}$ ).
The transmission voltage is doubled while the power and the line stay the same. By what factor does the power lost change?
Give two reasons why the grid uses alternating current.
List, in order, the voltage changes from a power station to a home in Rwanda, naming where each happens.
An engineer suggests sending power across Rwanda at 1 000 000 V "so the losses become tiny". Evaluate the suggestion.

Show answersHide answers

For a given power $P = V I$ , a high voltage means a small current, and the power wasted in the lines, $I^{2} R$ , is much smaller.
(a) $I = \frac{20 \times 10^{6}}{110 \times 10^{3}} \approx 182\ \text{A}$ , loss $\approx 182^{2} \times 4.0 \approx 132\ \text{kW}$ ; (b) $I \approx 90.9\ \text{A}$ , loss $\approx 33\ \text{kW}$ (a quarter).
$I = \frac{2000}{230} \approx 8.7\ \text{A}$ ; $P_{\text{loss}} = 8.7^{2} \times 0.50 \approx 38\ \text{W}$ .
$R = \frac{2.8 \times 10^{-8} \times 1.0 \times 10^{5}}{5.0 \times 10^{-4}} = 5.6\ \Omega$ .
The current halves, so the loss $I^{2} R$ falls to a quarter.
Transformers work only with a.c., so the voltage can be stepped up for transmission and down for use; a.c. generators are simple and efficient (and a.c. is easy to switch off safely).
Generator about 11 kV (typical) → step-up transformer at the power station to 110 kV or 220 kV → transmission lines → substation steps down to 30 kV or 15 kV → distribution lines → pole transformer steps down to 400 V and 230 V → homes.
The losses would indeed be very small, but the insulators, towers, clearances and land corridors would be very large and expensive, corona losses would grow, and the equipment would cost more than the energy saved, especially over the short distances in a small country like Rwanda. A balance between cost and losses is needed.

Key points

Power lost in a line: $P_{\text{loss}} = I^2 R = \left(\frac{P}{V}\right)^2 R$ , so high voltage means low loss.
Rwanda transmits at 110 kV and 220 kV (1,158 km of lines in June 2023), distributes at 30 kV and 15 kV, and supplies homes at 400 V and 230 V (REG).
The grid uses a.c. because transformers need it.
Very high voltages need costly insulation, towers and clearances, so a balance is chosen.

Sources and credits

20.4: Electric Power and Energy and 23.10: Transformers, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Consulted for the physics; no text was copied.
Rwandan grid facts: Transmission, Rwanda Energy Group (220 kV and 110 kV lines, 1,158 km at the end of June 2023); Rwanda completes a new transmission line to receive power from Rusumo hydropower plant, REG, 7 January 2023; Reticulation Standards, Version VI, REG, August 2020 (30 kV and 15 kV lines; 400 V and 230 V supply). The 11 kV generator voltage is a typical value, not a Rwandan figure.
Photo: The streets of Ruhengeri (8209006497).jpg by Carine06, Wikimedia Commons, licensed under CC BY-SA 2.0.
Video: GCSE Physics, National Grid, How we Generate and Transmit Electricity, Cognito, embedded from YouTube.

Written by Ganzaa: the lesson text, the grid diagram, the tables, worked examples and practice exercises.

1Why such dangerous voltages?

2Power lost in the lines

3From power station to home

4Why not an even higher voltage?

Why such dangerous voltages?

Power lost in the lines

From power station to home

Why not an even higher voltage?